As is defined through an equivalence relation, the canonical projection from to defines a topology ( the quotient topology ) and a differential structure on the projective line.
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A function, then the quotient topology on " Y " is the collection of subsets of " Y " that have open inverse images under " f ".
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Horizontal lines intersect the southern hemisphere in two antipodal points along the equator, either of which can be projected to the disk; it is understood that antipodal points on the boundary of the disk represent a single line . ( See quotient topology . ) So any set of lines through the origin can be pictured, almost perfectly, as a set of points in a disk.
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